Let $\Omega$ be a sample space.
A collection $\mathcal{D}$ of subsets of $\Omega$ is called a Dynkin sytem if
- $\Omega \in \mathcal{D}$
- $D \in \mathcal{D}$ implies $D^c \in \mathcal{D}$
- $D_i \in \mathcal{D}$, $i \geq 1$, pairwise disjoint implies $\bigcup_i D_i \in \mathcal{D}$.
A collection $\mathcal{C}$ of subsets of $\Omega$ is called a $\pi$ system if it is closed under intersection.
Now it seems to be quite clear that not every $\pi$-system can be a Dynkin system. But what about the other way round? Is every Dynkin system also a $\pi$-system?
By 2. and 3. it follows that $\bigcap_i D_i = (\bigcup_i D_i^c)^c \in \mathcal{D}$ when $D_i \in \mathcal{D}$ and the $D_i^c$ are pairwise disjoint. But I am not sure first what this means for the $D_i$ and whether one can make some other conclusion?
